rwg>Based on [Veit's] earlier analysis, I withdraw my claim about hypercubes.
The effective center varies (slightly) depending on what plane the centroid oscillates in. Wrong again. The moment of inertia of a cube about its center is orientation- independent. The center of oscillation, when hung from a vertex, always divides the major diagonal in proportion 11:7.
(Off list, Veit pointed out that the center of oscillation of a hoop swinging in its plane about a point on its rim is indeed its "south pole", as if all its mass were concentrated there. Absurd. But apparently irrefutable. Slightly less absurd: Turning an ordinary disk shaped pendulum 90 degrees to swing like a paddle shortens its period. (Assuming zero air resistance.) ---------- Doug Dodds at Symbolics used to send the following every year at this time. If you haven't seen one, check out "analemma" in Google images for one of those 365-exposure sky pix. There's even one for Mars!.--rwg --- Received: from ELEPHANT-BUTTE.SCRC.Symbolics.COM by RUSSIAN.SPA.Symbolics.COM via CHAOS with CHAOS-MAIL id 52312; Tue 8-Dec-87 13:52:55 PST Received: from YUKON.SCRC.Symbolics.COM by ELEPHANT-BUTTE.SCRC.Symbolics.COMvia CHAOS with CHAOS-MAIL id 244027; Tue 8-Dec-87 16:18:32 EST Received: from ANNISQUAM.SCRC.Symbolics.COM by YUKON.SCRC.Symbolics.COM via INTERNET with SMTP id 303771; 8 Dec 87 16:11:04 EST Date: Tue, 8 Dec 87 16:11 EST From: Doug Dodds <DODDS@YUKON.SCRC.Symbolics.COM> Subject: Rejoice, the light returneth! To: BBoard@YUKON.SCRC.Symbolics.COM, Fun@YUKON.SCRC.Symbolics.COM File-References: SCRC|Y:>Dodds>analemma.mss.newest, SCRC|Y:>Dodds>analemma.text.newest Message-ID: <19871208211112.0.DODDS@ANNISQUAM.SCRC.Symbolics.COM> The sunset today in Boston is at 16:11 EST, which is Plenty Early. But take heart, today's sunset is the earliest of the year! (Everywhere in the northern hemisphere, regardless of the actual sunset time.) The sunset starts to become later tomorrow, even though the shortest day is still two weeks off. If you haven't yet learned of this fascinating phenomenon, take a look at one of the referenced files. The first is formattable (m-X Format File); if formatting is not available to you, then print the second referenced file. Enjoy the increasing afternoon light! 0,, *** EOOH ***